Just hoping for some help with my practice exam. Any help on any of the questions would be greatly appreciated as my lecturer is absolutely shocking. Cheers.

(2)b

Let n ∈ N and let p be a prime. Show that if p | n then φ(np) = pφ(n).

Hint: consider the prime factorisation of n.

(17)

Show that the inverse of 5 modulo 101 is 5^99.

Use repeated squaring to simplify 5^99 (mod101)

Hence solve the equation 5x = 31(mod101)

(31)

Let P(R) be the set of all subsets of R; that is, P(R) = {X | X ⊆ R}. Let F be the set of all functions R → R. Define R : F → P(R) by R(f ) = {x ∈ R | f (x) = 0}. Prove

that R is surjective but not injective.

(36)b

In which of the following is (G, ∗) a semigroup? In which is it a group? Prove your answers.

G = N, a ∗ b = max{a, b}.

G = R \ { 1/2 }, a ∗ b = a − 2ab + b.

I know this is a lot of question but I'm really struggling and my test book doesn't have any examples. Thank you for all your time.